Commentary on Home page Step 4: The growth the universe is measured by entropy, a simple count of discrete states
Abstract:
A century of experimentation has identified a set of 61 elementary particles that form the foundation of the universe. Each one is either a fermion or a boson and can be represented by the 2D Hilbert space where we write the wave function: ψ = α|fermion⟩ + β|boson⟩, known in quantum computation as the qubit. Each vector in abstract Hilbert space corresponding to an elementary particle defines its genus as either fermion or boson, and contains further components that specify the species of each particle. These 61 particles are elements of the more complex structures that evolve in the Hilbert space of the universe. Elementary particle - Wikipedia, Fermion - Wikipedia, Boson - Wikipedia
Table Of Contents
1. Random action, the source of entropy
2. A brief history of entropy
3. The creative role of randomness: LaPlace’s Demon
4. At the smallest level the world is particulate
5. Hidden information and real eigenvalues
6. Infallible belief: a history of delusion
7. The “entropic overhead"
1. Random action, the source of entropy
Our task here is follow the creation of an enormously complex universe within an absolutely simple initial symmetry, the Aquinas–Einstein symetry. We measure our progress by increases in the entropy of the system, entropy being the measure of complexity.
The structureless Aquinas–Einstein symmetry that we imagine in step 1 as the eternal beginning of the universe starts with zero entropy and zero energy. In steps 2 and 3 we have populated it with a kinematic abstract Hilbert space and the mechanism of quantum mechanics.
We assume that Step 2 has produced a Hilbert space Cn with the superscript n representing a potential countably infinite number of orthogonal basis states. The number of discrete basis states produced by this random process is a measure of the entropy of the space. We may think of this space as as a text or a message and measure it in terms of Shannon’s mathematical theory of communication as having an alphabet A of n symbols an. Since the source of symbols at this point is random they can be considered equiprobable, and we may assign a probability of 1/n to each basis. Shannon’s entropy formula
H = Σn pn log2 pn
therefore simplifies to H = log2n.
This formula applies to all forms of symbolic communication including the messages passed to us from the physical world through the elementary particles described by quantum mechanics. Everything we see comes to us carried by photons; everything we hear through phonons; and our senses of touch, taste and smell are mediated by the interactions of particles in our bodies and the entities we are sensing.
Quantum mechanics divides these states into two orthogonal subspaces which may be represented by a qubit: |qubit⟩ = α|boson⟩ + β|fermion⟩. The states with symmetric statistics are bosons and those with antisymmetric statistics are fermions). Qubit - Wikipedia, Aleksandr Yakovlevich Khinchin (1960, 1998): The Mathematical Foundations of Quantum Statistics pp 80 sqq.
2. A brief history of entropy
The study of entropy began with the invention of heat engines. Early models of these machines were constructed by engineers like Thomas Newcomen who assembled arrangements of boilers, valves, pistons and linkages to use stream pressure to create mechanical energy. These engines were used to drive pumps to dewater coal mines. Engineers naturally sought to increase the efficiency of their engines to get as much work as possible out of a given amount of fuel.
After more than a century the theoretical answer to this problem was discovered by Sadi Carnot. He developed the abstract Carnot Cycle which imagined a heat engine working between a source of heating and source of cooling. At the beginning of the cycle compressed gas in the engine is heated and allowed to expand, doing some work on a piston. The gas is then cooled and compressed back to the original pressure, where it is heated once more and repeats the cycle. The cycle creates net mechanical energy because more work is done on the piston by the expanding hot gas that is required to compress the cold gas back to the beginning of the cycle. Nicolas Léonard Sadi Carnot - Wikipedia, Carnot cycle - Wikipedia
This process is theoretically reversible because the transfer of heat energy from the hot reservoir to the expanding gas is carried out at a constant temperature, as is the transfer of heat from the cold gas to the cold reservoir as it is compressed.
In theory such a cycle operating between a high temperature K1 and a low temperature K2 can produce an amount of mechanical energy proportional to (K1 - K2) / K1, where the temperatures are measured in Kelvin degrees above absolute zero. This figure is called the Carnot efficiency. This efficiency is ideal, since heat will not flow between between two substances at the same temperature. Carnot's Theorem (thermodynamics) - Wikipedia
Carnot published his book “Reflections on the Motive Power of Fire” in 1834 but his work attracted little attention. He died soon after in 1832.
He had, nevertheless opened the way to the science of thermodynamics. Two years after he died Émile Clapeyron drew attention to his work which led to the concepts of entropy, the quantity that is conserved in the Carnot cycle, and the second law of thermodynamics which states that entropy can only increase in an isolated system. Émile Clapeyron - Wikipedia
Ludwig Boltzmann studied entropy and the second law in detail and and contributed to the study of the relationship between heat and mechanical energy which shows that heat energy is simply another name for the kinetic energy of the vast numbers of microscopic particles in a macroscopic sample of any substance. He also studied the relationship between thermodynamic entropy and the statistical distribution of the kinetic energy of atoms and molecules. Entropy and ergodic theory have since became essential statistical components of information theory, which contributes to our understanding of the quantum mechanical creation of the universe. Ludwig Boltzmann - Wikipedia Boltzmann’s study of kinetic theory brought him to an impossible problem: the search for a mathematical proof of what we now know as his H-theorem. This proof became possible when John von Neumann applied David Hilbert’s interpretation of linear algebra to the study of physics and provided a consistent mathematical foundation for quantum mechanics. John von Neumann (1929_05_10): Proof of the Ergodic Theorem and the H-Theorem in Quantum Mechanics3. The creative role of randomness: LaPlace’s Demon
Many religious traditions attribute omnipotence, omniscience and detailed providence to the divinities who manage the universe. From this point of view every event, no matter how small, is under detailed divine control. The French mathematician Pierre-Simon Laplace proposed a quasi-divine being, since named LaPlace’s Demon, to study this possibility. He wrote:
We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past could be present before its eyes. Laplace's demon - Wikipedia
Such a deterministic world is doomed to repeat the same invariant history over and over again. Charles Darwin circumvented the sterility of this demonic history by realization that both variation and selection are essential inputs to biological evolution. He did not know about the role of genes and their random variation in reproduction but observed that living things do not breed precisely true, making the origin of new species possible over long periods of gradual change. Charles Darwin (1875, 1998): The Variation of Animals and Plants Under Domestication
return to toc4. At the smallest level the world is particulate
In Laplace’s time mathematics and logic were generally considered to be deterministic and Newton’s work on the solar system supported this view. Later, in 1859 Urbain Le Verrier found by precision measurements that the orbit of Mercury precessed by about 40 seconds of arc per century, motion that could no be explained by Newton’s work. Newton’s work was brought closer to reality by Einstein’s general theory which predicted this precession precisely. It has proven itself in the theory of black holes, gravitational waves and very high energy and precisely timed situations such as the Hulse-Taylor pulsar. It also serves to give us a picture of the universe as a whole which is supported by astronomical observations. Tests of general relativity - Wikipedia, Hulse-Taylor pulsar - Wikipedia
The general theory is a application of calculus to a spacetime-manifold defined by Einstein’s equivalence principle:
In Newton's law of gravity, the equivalence principle is viewed as an observation, an accident of nature encoded in empirical law. In Einstein's model of gravity, the equivalence principle is a consequence of physics. The same mass both causes and responds to gravity: mass causes spacetime curvature and then moves through spacetime according to that curvature. Equivalence principle - Wikipedia
Calculus was invented and applied by Isaac Newton in his description of the solar system because is provides a clear mathematical connection between velocity and acceleration in space and time which are presumed to be continuous.
This assumption is inconsistent with quantum mechanics at the microscopic scale. Planck discovered that he could not model black body radiation without assuming that it occurs in discrete quanta. Each carries energy in proportion to it frequency, E = hf where h is a new universal constant, the quantum of action. The value of h sets the scale of the microscopic world. The appearance of continuity, like the flow of water, is created by the vast numbers of particles involved. Planck constant - Wikipedia
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5. Hidden information and real eigenvalues
We represent a 2 dimensional Hilbert space with a qubit which we write as noted above as |qubit⟩ = α|boson⟩ + β|fermion⟩ with the proviso that α2 + β2 = 1 where α and β are complex numbers. This expression looks very similar to a vector of length 1 on a 2D plane, but from a quantum mechanical point of view this expression is statistical rather than geometrical. When we observe this qubit we see either a boson or a fermion, not a mixture of both. The only observable states of the qubit are those when α = 1 and β = 0 or vice-versa. The intermediate complex values of α and β take on meaning only when we make many observations on the same qubit. Then we find that we see bosons and fermions in the proportion of α2 to β2.
This suggests that there is “hidden information” in the qubit. In their book on quantum computation and quantum communication Nielsen and Chuang comment on this feature of quantum mechanics:
Understanding this hidden quantum information is a question that we grapple with for much of this book and which makes quantum mechanics a powerful took for information processing. Nielsen & Chuang (2016): Quantum Computation and Quantum Information, page 16.
This situation has been widely discussed under the general heading of “decoherence” and the discussion revolves around whether this feature is a defect in quantum mechanics which can be overcome by subtle technology, or whether it is simply the mathematical expression of the fact that the world is actually quantized. Much of the hope for quantum computation is built around the idea that we can somehow get behind this quantization to the apparent continuity of the “interior” of the qubit (or other quantum expressions in higher dimensional Hilbert spaces). The idea seems to be that if we could get inside the qubit we could use it as a continuous perfect analogue computer and solve an infinite continuum of classically incomputable problems “within” the qubit, overcoming the inherent limitation of the Turing machine which can only, in theory, solve a countable infinity of problems.
An example of this hope is Shor’s algorithm for factorization which might be able to crack codes whose security depends on the difficulty of factorizing large numbers. This facility may be a long way off, since it requires a large quantum computer. As Wikipedia tells us
Shor's algorithm shows that factoring integers can be done with a polynomial complexity circuit on an ideal quantum computer. Thus, it might be feasible to defeat RSA by constructing a large enough quantum computer. This was a powerful motivator for the design and construction of quantum computers, and for the study of new quantum-computer algorithms. It has also facilitated research on new cryptosystems that are secure from quantum computers, collectively called post-quantum cryptography (PQC). Shor's algorithm - Wikipedia
For the time being we must accept the normal protocol of quantum observation which, in the case of the qubit we are using, provided us, on repeated observation, with fermions and bosons in proportion to the ratio of α2 to β2.
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Page 6. Infallible belief: a history of delusion
A central issue for many traditional theologies is the existence of evil. The question is whether an omnipotent, omniscient creator wth the happiness and pleasure of humanity at heart could allow, or even cause evil in the world, particularly war, disease, famine and in particular, hatred of various groups of people for one another.
The story of creation on this site is developed in contrast to the Doctrine of the Roman Catholic Church. This Church has gradually developed its body of doctrine over about three millennia since the people of Israel left Egypt and moved to Palestine. Its most recent major revision was the Second Vatican Council in 1965. The Catechism of the Catholic Church is a new extensive compendium of Catholic belief published in English in 1994 with revised version in 2000 based on the 1997 Latin revision. The Catechism comprises 2865 paragraphs of varying length with very extensive references to its historical sources, particularly the Catholic Bible and the Fathers of the Church. Libreria Editrice Vaticana: Catechism of the Catholic Church (Second Edition)
The reason for the existence of this Roman Catholic Church is the Fall, a mysterious act of disobedience recorded in Genesis, the first book of Moses. The Catechism records the Catholic interpretation of this event in 8 short paragraphs:
return to toc§413: “God did not make death, and he does not delight in the death of the living [. . .] . It was through the devil’s envy that death entered the world” (Wis 1:13; 2:24).
§414: Satan or the devil and the other demons are fallen angels who have freely refused to serve God and his plan. Their choice against God is definitive. They try to associate man in their revolt against God.
§415: “Although set by God in a state of rectitude, man, enticed by the evil one, abused his freedom at the very start of history. He lifted himself up against God and sought to attain his goal apart from him”.
§416: By his sin Adam, as the first man, lost the original holiness and justice he had received from God, not only for himself but for all human beings.
§417: Adam and Eve transmitted to their descendants human nature wounded by their own first sin and hence deprived of original holiness and justice; this deprivation is called “original sin.”
§418: As a result of original sin, human nature is weakened in its powers; subject to ignorance, suffering, and the domination of death; and inclined to sin (This inclination is called “concupiscence”).
§419: “We therefore hold, with the Council of Trent, that original sin is transmitted with human nature, ‘by propagation, not by imitation’ and that it is... ‘proper to each’” (Paul VI, CPG § 16).
§420: The victory that Christ won over sin has given us greater blessings than those which sin had taken from us: “where sin increased, grace abounded all the more” (Rom 5:20).
§421: Christians believe that “the world has been established and kept in being by the Creator’s love; has fallen into slavery to sin but has been set free by Christ, crucified and risen to break the power of the evil one...”
page 7: The “entropic overhead"
Traditional Christian theology gives God absolute control of all events in the world with the exception of intelligent beings such as angels and humans. This exception makes it possible for some of creatures to act against the will of the omnipotent god. This opened the way for the whole story of salvation involving the miraculous conception of Jesus of Nazareth the Son of God, his foreshadowing of the Church and his crucifixion to win back the favour of the Creator and ultimatly the establishment of the Christian Church to distribute the grace of salvation. All of this may be conceived as a rather wasteful divine plan premissed on the need of the divinity to create free creatures to worship them.
The actual process of evolution may also be seen as wasteful without any premeditation by an omnipotent and omniscient being, since we see, particularly among plants and more primitive organisms, very large numbers of individuals being reproduced with very little chance of survival. From the point of view of life, we see an enormous creation and destruction of entropy.
What we have learnt about the world of physics, however, points to a rigid law of conservation of energy which operates as a major constraint on the evolution of the physical world. Energy it would seem is neither created nor destroyed. We now need to explore how energy is created and maintained within the initial symmetry.
Copyright:
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Notes and references
Further reading
Books
Darwin (1875, 1998), Charles, and Harriet Ritvo (Introduction), The Variation of Animals and Plants Under Domestication (Foundations of Natural History), Johns Hopkins University Press 1875, 1998 ' "The Variation, with its thousands of hard-won observations of the facts of variation in domesticated species, is a frustrating, but worthwhile read, for it reveals the Darwin we rarely see -- the embattled Darwin, struggling to keep his project on the road. Sometimes he seems on the verge of being overwhelmed by the problems he is dealing with, but then a curious fact of natural history will engage him (the webbing between water gun-dogs' toes, the absurdly short beak of the pouter pigeon) and his determination to make sense of it rekindles. As he disarmingly declares, 'the whole subject of inheritance is wonderful.'.
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Khinchin (1960, 1998), Aleksandr Yakovlevich, The Mathematical Foundations of Quantum Statistics, Dover 1998 'In the area of quantum statistics, I show that a rigorous mathematical basis of the computational formulas of statistical physics . . . may be obtained from an elementary application of the well-developed limit theorems of the theory of probability.'
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Nielsen (2016), Michael A., and Isaac L Chuang, Quantum Computation and Quantum Information, Cambridge University Press 2016 Review: A rigorous, comprehensive text on quantum information is timely. The study of quantum information and computation represents a particularly direct route to understanding quantum mechanics. Unlike the traditional route to quantum mechanics via Schroedinger's equation and the hydrogen atom, the study of quantum information requires no calculus, merely a knowledge of complex numbers and matrix multiplication. In addition, quantum information processing gives direct access to the traditionally advanced topics of measurement of quantum systems and decoherence.' Seth Lloyd, Department of Quantum Mechanical Engineering, MIT, Nature 6876: vol 416 page 19, 7 March 2002.
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Links
, Hulse-Taylor pulsar - Wikipedia, the free encyclopedia, The Hulse–Taylor pulsar . . . is a binary star system composed of a neutron star and a pulsar which orbit around their common center of mass. It is the first binary pulsar ever discovered. The pulsar was discovered by Russell Alan Hulse and Joseph Hooton Taylor Jr., of the University of Massachusetts Amherst in 1974. . .. The orbit has decayed since the binary system was initially discovered, in precise agreement with the loss of energy due to gravitational waves described by Albert Einstein's general theory of relativity. The ratio of observed to predicted rate of orbital decay is calculated to be 0.997 ± 0.002. The total power of the gravitational waves emitted by this system presently is calculated to be 7.35 × 10^24 watts. For comparison, this is 1.9% of the power radiated in light by the Sun.' back |
Albert Einstein (1905), On the Electrodynamics of Moving Bodiesx, An english translation of the paper that founded Special relativity. 'Examples of this sort, [in the contemporary application of Maxwell's electrodynamics to moving bodies] together with the unsuccessful attempts to discover any motion of the earth relatively to the ``light medium,'' suggest that the phenomena of electrodynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest. They suggest rather that, as has already been shown to the first order of small quantities, the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good.' back |
Alessandro Volta - Wikipedia, Alessandro Volta - Wikipedia, the free encyclopedia, 'Alessandro Giuseppe Antonio Anastasio Volta (18 February 1745 – 5 March 1827) was an Italian physicist and chemist who was a pioneer of electricity and power who is credited as the inventor of the electric battery and the discoverer of methane. He invented the voltaic pile in 1799, and reported the results of his experiments in 1800 in a two-part letter to the president of the Royal Society.' back |
Boltzmann equation - Wikipedia, Boltzmann equation - Wikipedia, the free equation, ' The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium; it was devised by Ludwig Boltzmann in 1872. The classic example of such a system is a fluid with temperature gradients in space causing heat to flow from hotter regions to colder ones, by the random but biased transport of the particles making up that fluid. In the modern literature the term Boltzmann equation is often used in a more general sense, referring to any kinetic equation that describes the change of a macroscopic quantity in a thermodynamic system, such as energy, charge or particle number. The equation arises not by analyzing the individual positions and momenta of each particle in the fluid but rather by considering a probability distribution for the position and momentum of a typical particle—that is, the probability that the particle occupies a given very small region of space (mathematically the volume element d.3r centered at the position r and has momentum nearly equal to a given momentum vector p (thus occupying a very small region of momentum space d3p at an instant of time. The Boltzmann equation can be used to determine how physical quantities change, such as heat energy and momentum, when a fluid is in transport. One may also derive other properties characteristic to fluids such as viscosity, thermal conductivity, and electrical conductivity (by treating the charge carriers in a material as a gas). See also convection–diffusion equation.' The equation is a nonlinear integro-differential equation, and the unknown function in the equation is a probability density function in six-dimensional space of a particle position and momentum. The problem of existence and uniqueness of solutions is still not fully resolved, but some recent results are quite promising. back |
Boson - Wikipedia, Boson - Wikipedia, the free encyclopedia, 'In particle physics, bosons are particles with an integer spin, as opposed to fermions which have half-integer spin. From a behaviour point of view, fermions are particles that obey the Fermi-Dirac statistics while bosons are particles that obey the Bose-Einstein statistics. They may be either elementary, like the photon, or composite, as mesons. All force carrier particles are bosons. They are named after Satyendra Nath Bose. In contrast to fermions, several bosons can occupy the same quantum state. Thus, bosons with the same energy can occupy the same place in space.' back |
Carnot cycle - Wikipedia, Carnot cycle - Wikipedia, the free encyclopedia, ' The Carnot cycle is a theoretical thermodynamic cycle proposed by Nicolas Léonard Sadi Carnot in 1824 and expanded by Benoit Paul Émile Clapeyron in the 1830s and 40s. It can be shown that it is the most efficient cycle for converting a given amount of thermal energy into work, or conversely, creating a temperature difference (e.g. refrigeration) by doing a given amount of work.' back |
Carnot's Theorem (thermodynamics) - Wikipedia, Carnot's Theorem (thermodynamics) - Wikipedia, the free ncyclopedia, ' Carnot's theorem, also called Carnot's rule or Carnot's law, is a principle of thermodynamics developed by Nicolas Léonard Sadi Carnot in 1824 that specifies limits on the maximum efficiency that any heat engine can obtain.' back |
Catechism of the Catholic Church, Table of Contents, Prologue (#Sect;§ 1 - 25; Part I: The Profession of Faith (§§26 - 1065 Part II: The Celebration of th Christian Mystery (§§1066 - 1690 Part III: Life in Christ: (§§1691 - 2557) Part IV: Christian Prayer: (§§2558 - 2865) back |
Catechism of the Catholic Church - Wikipedia, Catechism of the Catholic Church - Wikipedia, the free encyclopedia, 'The Catechism of the Catholic Church (Latin: Catechismus Catholicae Ecclesiae; commonly called the Catechism or the CCC) is a reference work that summarizes the Catholic Church's doctrine. It was promulgated by Pope John Paul II in 1992 as a reference for the development of local catechisms, directed primarily to those (in the church) responsible for catechesis and offered as "useful reading for all other Christian faithful". It has been translated into and published in more than twenty languages worldwide. |
Claude E Shannon (1948), A Mathematical Theory of Communication, '' The recent development of various methods of modulation such as PCM and PPM which exchange T bandwidth for signal-to-noise ratio has intensified the interest in a general theory of communication. A basis for such a theory is contained in the important papers of Nyquist1 and Hartley2 on this subject. In the present paper we will extend the theory to include a number of new factors, in particular the effect of noise in the channel, and the savings possible due to the statistical structure of the original message and due to the nature of the final destination of the information. |
Claude Shannon (1949), Communication in the presence of noise, 'A method is developed for representing any communication system geometrically. Messages and the corresponding signals are points in two “function spaces,” and the modulation process is a mapping of one space into the other. Using this representation, a number of results in communication theory are deduced concerning expansion and compression of bandwidth and the threshold effect. Formulas are found for the maximum rate of transmission of binary digits over a system when the signal is perturbed by various types of noise. Some of the properties of “ideal” systems which transmit at this maximum rate are discussed. The equivalent number of binary digits per second for certain information sources is calculated.' [C. E. Shannon , “Communication in the presence of noise,” Proc. IRE, vol. 37, pp. 10–21, Jan. 1949.] back |
Elementary particle - Wikipedia, Elementary particle - Wikipedia, the free encyclopedia, ' In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include the fundamental fermions (quarks, leptons, antiquarks, and antileptons), which generally are "matter particles" and "antimatter particles", as well as the fundamental bosons (gauge bosons and the Higgs boson), which generally are "force particles" that mediate interactions among fermions. A particle containing two or more elementary particles is a composite particle.' back |
Émile Clapeyron - Wikipedia, Émile Clapeyron - Wikipedia, the free encyclopedia, ' Benoît Paul Émile Clapeyron (26 January 1799 – 28 January 1864) was a French engineer and physicist, one of the founders of thermodynamics. He is known for developing the ideal gas law. [. . .] |
Equivalence principle - Wikipedia, Equivalence principle - Wikipedia, the free encyclopedia, ' In the physics of general relativity, the equivalence principle is any of several related concepts dealing with the equivalence of gravitational and inertial mass, and to Albert Einstein's assertion that the gravitational "force" as experienced locally while standing on a massive body (such as the Earth) is actually the same as the pseudo-force experienced by an observer in a non-inertial (accelerated) frame of reference. [. . .] |
Fermion - Wikipedia, Fermion - Wikipedia, the free encyclopedia, 'In particle physics, fermions are particles with a half-integer spin, such as protons and electrons. They obey the Fermi-Dirac statistics and are named after Enrico Fermi. In the Standard Model there are two types of elementary fermions: quarks and leptons. . . . In contrast to bosons, only one fermion can occupy a quantum state at a given time (they obey the Pauli Exclusion Principle). Thus, if more than one fermion occupies the same place in space, the properties of each fermion (e.g. its spin) must be different from the rest. Therefore fermions are usually related with matter while bosons are related with radiation, though the separation between the two is not clear in quantum physics. back |
General relativity - Wikipedia, General relativity - Wikipedia, the free encyclopedia, 'General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics. General relativity generalises special relativity and Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or spacetime. In particular, the curvature of spacetime is directly related to the four-momentum (mass-energy and linear momentum) of whatever matter and radiation are present. The relation is specified by the Einstein field equations, a system of partial differential equations.' back |
Heinrich Hertz - Wikipedia, Heinrich Hertz - Wikipedia, the free encyclopedia, ' Heinrich Rudolf Hertz (22 February 1857 – 1 January 1894) was a German physicist who first conclusively proved the existence of the electromagnetic waves predicted by James Clerk Maxwell's equations of electromagnetism. The unit of frequency, cycle per second, was named the "hertz" in his honor.' back |
Hylomorphism - Wikipedia, Hylomorphism - Wikipedia, the free encyclopedia, 'Hylomorphism (Greek ὑλο- hylo-, "wood, matter" + -morphism < Greek μορφή, morphē, "form") is a philosophical theory developed by Aristotle, which analyzes substance into matter and form. Substances are conceived of as compounds of form and matter.' back |
John von Neumann (1929_05_10), Proof of the Ergodic Theorjjem and the H-Theorem in Quantum Mechanics, ' Abstract:
It is shown how to resolve the apparent contradiction between the macroscopic approach of phase space and the validity of the uncertainty relations. The main notions of statistical mechanics are re-interpreted in a quantum-mechanical way, the ergodic theorem and the H-theorem are formulated and proven (without “assumptions of disorder”), followed by a discussion of the physical meaning of the mathematical conditions characterizing their domain of validity. [. . .] |
Kinetic theory of gases - Wikipedia, Kinetic theory of gases - Wikipedia, the free encyclopedia, 'The kinetic theory of gases is a simple, historically significant classical model of the thermodynamic behavior of gases, with which many principal concepts of thermodynamics were established. The model describes a gas as a large number of identical submicroscopic particles (atoms or molecules), all of which are in constant, rapid, random motion. Their size is assumed to be much smaller than the average distance between the particles. The particles undergo random elastic collisions between themselves and with the enclosing walls of the container. The basic version of the model describes the ideal gas, and considers no other interactions between the particles.' back |
Laplace's demon - Wikipedia, Laplace's demon - Wikipedia, the free encyclopedia, ' In the history of science, Laplace's demon was a notable published articulation of causal determinism on a scientific basis by Pierre-Simon Laplace in 1814. According to determinism, if someone (the demon) knows the precise location and momentum of every particle in the universe, their past and future values for any given time are entailed; they can be calculated from the laws of classical mechanics. |
Libreria Editrice Vaticana, Catechism of the Catholic Church (Second Edition), back |
Ludwig Boltzmann - Wikipedia, Ludwig Boltzmann - Wikipedia, the free encyclopedia, Ludwig Eduard Boltzmann (February 20, 1844 – September 5, 1906) was an Austrian physicist and philosopher whose greatest achievement was in the development of statistical mechanics, which explains and predicts how the properties of atoms (such as mass, charge, and structure) determine the physical properties of matter (such as viscosity, thermal conductivity, and diffusion). back |
Maxwell's equations - Wikipedia, Maxwell's equations - Wikipedia, the free encyclopedia, ' Maxwell's equations are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar etc. They describe how electric and magnetic fields are generated by charges, currents, and changes of the fields.' back |
Michael Faraday - Wikipedia, Michael Faraday - Wikipedia, the free encyclopedia, ' Michael Faraday FRS September 1791 – 25 August 1867) was an English scientist who contributed to the study of electromagnetism and electrochemistry. His main discoveries include the principles underlying electromagnetic induction, diamagnetism and electrolysis. Although Faraday received little formal education, he was one of the most influential scientists in history. It was by his research on the magnetic field around a conductor carrying a direct current that Faraday established the concept of the electromagnetic field in physics. . . . His inventions of electromagnetic rotary devices formed the foundation of electric motor technology, and it was largely due to his efforts that electricity became practical for use in technology.' back |
Nicolas Léonard Sadi Carnot - Wikipedia, Nicolas Léonard Sadi Carnot - Wikipedia, the free encyclopedia, ' Sous-lieutenant Nicolas Léonard Sadi Carnot (1 June 1796 – 24 August 1832) was a French mechanical engineer in the French Army, military scientist and physicist, and often described as the "father of thermodynamics." Like Copernicus, he published only one book, the Reflections on the Motive Power of Fire (Paris, 1824), in which he expressed, at the age of 27 years, the first successful theory of the maximum efficiency of heat engines. In this work, he laid the foundations of an entirely new discipline: thermodynamics.' back |
Planck constant - Wikipedia, Planck constant - Wikipedia, the free encyclopedia, ' Since energy and mass are equivalent, the Planck constant also relates mass to frequency. By 2017, the Planck constant had been measured with sufficient accuracy in terms of the SI base units, that it was central to replacing the metal cylinder, called the International Prototype of the Kilogram (IPK), that had defined the kilogram since 1889. [. . .] For this new definition of the kilogram, the Planck constant, as defined by the ISO standard, was set to 6.626 070 150 × 10-34 J⋅s exactly. ' back |
Qubit - Wikipedia, Qubit - Wikipedia, the free encyclopedia, 'A quantum bit, or qubit . . . is a unit of quantum information. That information is described by a state vector in a two-level quantum mechanical system which is formally equivalent to a two-dimensional vector space over the complex numbers. Benjamin Schumacher discovered a way of interpreting quantum states as information. He came up with a way of compressing the information in a state, and storing the information on a smaller number of states. This is now known as Schumacher compression. In the acknowledgments of his paper (Phys. Rev. A 51, 2738), Schumacher states that the term qubit was invented in jest, during his conversations with Bill Wootters.' back |
Shor's algorithm - Wikipedia, Shor's algorithm - Wikipedia, the free encyclopedia, ' Shor's algorithm is a polynomial-time quantum computer algorithm for integer factorization. Informally, it solves the following problem: Given an integer N, find its prime factors. It was invented in 1994 by the American mathematician Peter Shor. . . . If a quantum computer with a sufficient number of qubits could operate without succumbing to quantum noise and other quantum-decoherence phenomena, then Shor's algorithm could be used to break public-key cryptography schemes, such as the widely used RSA scheme. RSA is based on the assumption that factoring large integers is computationally intractable.' back |
Tests of general relativity - Wikipedia, Tests of general relativity - Wikipedia, the free encyclopedia, ' Tests of general relativity serve to establish observational evidence for the theory of general relativity. The first three tests, proposed by Albert Einstein in 1915, concerned the "anomalous" precession of the perihelion of Mercury, the bending of light in gravitational fields, and the gravitational redshift. The precession of Mercury was already known; experiments showing light bending in accordance with the predictions of general relativity were performed in 1919, with increasingly precise measurements made in subsequent tests; and scientists claimed to have measured the gravitational redshift in 1925, although measurements sensitive enough to actually confirm the theory were not made until 1954. A more accurate program starting in 1959 tested general relativity in the weak gravitational field limit, severely limiting possible deviations from the theory. [. . .] |